Complex Natural Logarithm/Examples/i
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Example of Complex Natural Logarithm
- $\ln \paren i = \paren {4 k + 1} \dfrac {\pi i} 2$
for all $k \in \Z$.
Proof
\(\ds i\) | \(=\) | \(\ds \exp \paren {\dfrac {i \pi} 2}\) | Polar Form of $i$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \ln \paren i\) | \(=\) | \(\ds \ln \paren {\exp \paren {\dfrac {i \pi} 2 + 2 k \pi i} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {i \pi + 4 k \pi i} 2\) | Definition of Complex Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {4 k + 1} \dfrac {\pi i} 2\) |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.6$. The Logarithm: Examples: $\text {(ii)}$